Time Delays, also known as time lags or transport delays, often arise in control processes. Long delays (relative to the dominant time constant of the process) in control processes tend to limit the performance of control systems. In contrast, short delays (relative to the dominant time constant of the process) are also common in control processes and like long delays tend to limit the performance of a control system.
Numerous attempts have been undertaken to extend classical and modern control techniques to accommodate time delays in linear systems. However, the results of these efforts offer limited support for linear systems with delays. Moreover, those solutions capable of handling delay differential equations (DDEs) are considered restrictive and burdensome for use to design control systems because they do not work well with standard linear analysis tools such as the Control System Toolbox available from The MathWorks®, of Natick, Mass.
One known theory well suited for use in describing a control processes is linear time invariant (LTI) system theory. LTI systems form a large and useful class of systems well suited to describe, amongst others, communication channels, manufacturing systems, airplanes, and economic systems. LTI systems are dynamic systems having the following properties. First, the behavior of an LTI system does not change with time, that is, LTI systems are time invariant. Second, the input/output relationship of an LTI system is linear, that is, the system obeys the superposition principle. As such, LTI systems are often fundamental building blocks for theoretical and applied control engineering applications.
An LTI system is describable according to the following first order state space formulas:
                                          ⅆ            x                                ⅆ            t                          =                              Ax            ⁡                          (              t              )                                +                      Bu            ⁡                          (              t              )                                                          (                  Eq          .                                          ⁢          1                )            y(t)=Cx(t)+Du(t)  (Eq. 2)
The input to the LTI system u(t) and the output of the LTI system y(t) are considered vector valued signals. The state vector of the LTI system is represented by x(t). In the Laplace domain, the input/output transfer function H(s) of this LTI system is definable by the following formula:H(s)=D+C(sI−A)−1B  (Eq. 3)
Equations 1-3 define a class of LTI systems often used to conduct linear analysis and design of control systems and processes. Nonetheless, such classes of LTI systems provide little if any support for time delays. It is well known that time delays are common in actual systems, however, it is also common for control system modeling or analysis tools to lack a general framework for representing, manipulating, and analyzing LTI systems with delays. Consequently, it is often difficult to model, analyze, and design systems, such as control systems and processes with delays. Therefore, there exists a need for a framework for manipulating and analyzing LTI systems with delays.